Abstract
In the paper we present a proof of the local criterion for crystalline structures which generalizes the local criterion for regular systems. A Delone set is called a crystal if it is invariant with respect to a crystallographic group. Locally antipodal Delone sets, i.e. those in which all 2R-clusters are centrally symmetrical, are considered and we prove that they have crystalline structure. Moreover, if in a locally antipodal set all 2R-clusters are the same, then the set is a regular system, i.e. a Delone set whose symmetry group operates transitively on the set.
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Work is supported by the RNF grant 14-11-00414.
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Notes
- 1.
The concept of a ‘periodic’ crystal as the union of several crystallographic point orbits goes back to Fedorov. After Shechtman’s discovery of ‘aperiodic’ quasicrystals, the International Union for Crystallography has extended the concept of crystal by including periodic (in Fedorov’s sense) as well as aperiodic (in Shechtman’s sense) crystals. Though a search for local conditions for aperiodic crystals remains extremely interesting and unsolved problem, throughout the paper we will mean under ‘crystal’ periodic crystals only.
- 2.
In September 2016 the author presented this construction in his talk at the American Institute of Mathematics on a workshop “Soft Packings, Nested Clusters, and Condensed Matter”. The construction raised in frames of the workshop a fruitful discussion on possible extending this example for any dimension. The discussion led a group of the workshop’s participants to the following result: For any dimension d and \(\varepsilon >0\) there is a non-regular Delone set with \(N(d2R-\varepsilon )=1\).
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Acknowledgements
The author thanks Nikolay Andreev (Moscow) for making drawings and Andrey Ordine (Toronto) for his help in editing the English text. The author is very grateful to the anonymous reviewer for having made numerous significant comments that helped to improve this paper.
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Dolbilin, N. (2018). Delone Sets: Local Identity and Global Symmetry. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_6
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